1901 — 2 .] Dr Muir on the Theory of Orthogonanls. 
271 
£ — + V\V + z \ z + 
rj = x 2 x + y 2 y + z 2 z + 
l = + y z y + z z z + 
In the fourth place, if we take the second of these derived sets of 
■equations, multiplication by x , y , z , . . . respectively, followed 
by addition, gives 
A xx x 2 + A w y 2 + • • ■ + 2 A XIJ xy + • • • • 
= s,£ 2 + s 2 f + s s £ 2 + ... 
With these results before him Cauchy is led to formulate the 
following proposition previously given “dans le dernier volume 
-des Memoires de V Academie des Sciences ” : — 
“Theoiteme II. Etant donnee une fonction homogene et du 
second degre de plusieurs variables x , y , z , ... , on peut 
toujours leur substituer d’autres variables £ , rj , £ , ... 
liees a x, y , z , .... par des equations lineaires tellement 
clioisies que la somme des carres de x , y , z , ... soit 
equivalente a la somme des carres de £, rj , l, ... , et 
que la fonction donnee de x , y , z , ... se transforme en 
une fonction £ , rj , £ , . : . . homogene et du second degre, 
mais qui renferme seulement les carres de £ , 77 , £ , ... ” 
The validity of this rests on the supposition that the equation 
R, = 0 has all its roots unequal ; but Cauchy is careful to point out 
that even if this were not the case, the requisite inequality 
■could be brought about by giving an infinitely small increment e 
to one of the coefficients A xx , A xy , ; and as e could be made 
to approach indefinitely near to zero without the theorem ceasing 
to be valid, the validity would remain even at the limit. 
After a reference to the special case of three variables, the 
paper closes with the announcement that Sturm had arrived 
independently at the theorems marked I. and II., and had offered 
his paper on the subject to the Academy on the same day as 
Cauchy’s.* 
* A short account of Cauchy’s memoir is given in the Bulletin des Sciences 
■Math., xii. (1829), pp. 301-303, by C. S(turm), who says, “M. Cauchy a bien 
voulu observer, en terminant son article, que j’etais parvenu, de mon cote, a 
